John,
Found this quote somewhere.......seems consistent when you look for 16f628a bootloader.

Bootloaders don't do serial port programming in the traditional sense. The
whole point of a bootloader is that the chip can program itself, and the
bootloader then can choose the appropriate interface for getting data into
the chip to be programmed. So you don't need to know the traditional
Micorchip serial programming algorithm, but the algorithm for self
programming the chip, which I remind you that the 16F628A cannot do.

The simplest solution is to upgrade your 16F628A to a 16F88. They are
virtually pin for pin compatible and the 16F88 can rewrite its own program
memory, making it bootloadable. Other 18 pin PIC in the family that can do
that are the 16F818/819.

And also this............

The only bootload-like thing you can do on a 16F628A is to store an
interpreter in the Flash and the program-to-be-interpreted in EEPROM.

So maybe this will help your programmer to see how the original was done. (Just a thought if you go this way in future)

Cheers....Jeff